First you need to get rid of all the white from the lineart layer. There are multiple ways of doing this and people have already written much better tutorials than me on how to do it so I'll just link to them.
I use this method: http://www.howtogeek.com/113365/how-to-remove-the-background-from-a-drawing-or-lineart/
This works too and is even simpler (steps 1-5): http://cgcookie.com/concept/2012/03/16/tip-turning-scanned-pencil-sketches-into-digital-paintings/
Done? OK! You should have a layer something like this:
(Sorry, this is one I drew digitally because I have no inked-in-real-life lineart on my computer to work with. It's the same principle though, you'll have black/greyscale lineart on a transparent layer with no white on it.)
Next, create a new layer (Layer > New) above the lineart.
Then hover "between" the two layers (I arrowed it in the screenshot so you can see what I mean) in the layer palette and hold down the alt key. The icon changes to two little circles with a line between. Click!
You should now see the upper layer has jumped inside the lower one, as below:
Now with the upper layer selected, get your paintbrush and start colouring over the lines you want coloured. The upper layer is automatically clipped to the one beneath, like this:
You can be as messy as you want when colouring this in! You only need to be a little careful on the areas where the two colours meet. You can also fill it with the paint bucket, drop a whole other texture or something in there or fill it with a gradient - try it out! And if you want to make a new layer for each colour, you can do that too, just make a new one and hold alt again over the top layer, or drag it in between the two depending on what order you want them in.
Then you can just make new your new layers underneath the lineart layer and colour them as you normally would!